Logic in Programming Logic is extremely important in both the hardware and software of computing. Here we will begin with the software aspects of logic which are involved in programming. Later we will briefly show some hardware aspects of gates involved in computing architecture. Logic involves conditions in virtually all Choice and Loop constructs (If, and While forms). Logic also occurs in assertions, preconditions, post conditions, invariants and even comments. The boolean (or logical) type is binary; it has only two values, true and false in angular boxes: true false Logical boxes (variables) include such examples as: male, female, tall, old, done, isIsosceles which often must be clarified, perhaps by comments. For example, "old" could be defined as as any age over 21 (or 12, 30, 100, whatever). Declarations of logical types specify boxes having only the two truth values. They have a form which begins with "boolean" followed by any number of variables separated by commas and end with a semicolon. Examples are: boolean male, female; boolean old; // age over 21 boolean tall; // height over or equal to 72 inches Assignment of truth values to boolean boxes (variables) can be done simply as: male = true; old = false; 1 tall = old ; Arithmetic Relations Arithmetic relations often occur in logical conditions; the relations compare two quantities of the same type (such as ints here). (a < b) which reads "a is less than b" (c > d) which reads "c is greater than d" or "c is more than d" (e <= f) which reads "e is less than or equal to f" (g >= h) which reads "g is greater than or equal to h" (i == j) which reads "i is equal to j" (j != k) which reads "j is not equal to k" Alternate or equivalent ways are possible to express conditions: (p < q) is equivalent to (q > p) for example (age < 12) is equivalent to (12 > age) since they both are true for values 11, 10, 9, 8, .. etc and both are false for values 12 and 13, 14, 15, …, etc Similarly, other equivalent conditions are: (x <= y) is equivalent to (y >= x) as in the example: (7 <= sum) is equivalent to (sum >= 7) Notice especially that (a < b) is not the opposite or complement of (a > b). 2 Complements are treated shortly. Assignment may also involve comparison with relations such as: over21 = (age > 21); where over21 is true for age 22, 23, 24 .. etc and is false for ages 21,20, 19 .. etc. Other examples of logical assignment are: tied = (visitorScore == homeScore); error = (age < 0); proper = (percent <= 100); tall = (height >= 72); // inches error2 = (denominator == 0); Use of logical terms can be done in a number of ways. For example, legal = (over21 == true); can be done simpler and shorter with legal = over21; Equivalent Control Forms: (optional at first) Logical assignments can also be done as imperative or control-oriented forms. For example the assignment over21 = (age > 21); can be written as the Choice form: -- PseudoCode // Java Code If age > 21 if (age > 21) { Set over21 = true over21 = true; Else } else { Set over21 = false over21 = false; EndIf }//end if 3 Complements (Opposites, negatives, inverses) Complements are logical opposites; when one is true the complement is false. Examples of complements include male, female also tall, short and old, young. Complements are expressed with the logical operator "!" which reads "not". The complement, or Not, is unary, it acts on the one condition that follows it. If the boolean box b is true then the complement ! b is false; if b is false then ! b is true, as show in the following truth table, bVal ! bVal b !b false true and a shorter version F T true false T F For example, consider the assignment: small = ! tall ; If tall is true then small is false, and if tall is false then small is true. Other example assignments follow: female= ! male; young = ! (age > 12); Complements can involve arithmetic relations: (a < b) is the complement of (a >= b) (a > b) is the complement of (a <= b) (a == b) is the complement of (a != b) The above condition for young can be written without the not operator as: !(age > 12) is (age <= 12) Similarly: !(age <= 21) is (age > 21) 4 Logical Binary Operations (And and Or) Operations on logical or boolean boxes include two binary operators (And and Or); The first letter of And and Or may be capitalized to avoid confusion, as in the line above. And (also called "andAlso" or conjunction, symbolized by &) Or (also called "eitherOr" or disjunction, symbolized by |) Binary operations (Or, And) operate on two operands; the operator is between the two operands (called infix). And is a logical operation often called the conjunction. The And denoted (p && q) is true when p is true and q is true. Examples of the use of this And follow. increasing = (x < y) & (y < z); equilateral = (s1 == s2) & (s1 == s3); isInRange = (percent >= 0) & (percent <= 100); isEligible = over21 & isEmployed; Truth table of the logical And operator is shown below, including a shorter form; The And of two operands is true only in one case, when both p and q are true. first second first & second p q p & q false false false F F F false true false F T F true false false T F F true true true T T T 5 Or is a logical operation often called the disjunction, eitherOr, or inclusive-Or; The Or denoted (p | q) is true when either p or q or both are true. Examples of the use of the Or follow, in Java. winpoint = (sum == 7) | (sum = 11); error = (percent < 0) | (percent > 100); playBall = (inning <= 9) | (score1 == score2); isosceles = (a == b) | (b == c) | (c == a); Truth table of the logical Or operator is shown below. The Or of two operands is true in 3 cases, when either p or q are true. The Or is false only in one case, when both p and q are false (or neither is true). Or Xor p q p | q p q p ^ q F F F F F F F T T F T T T F T T F T T T T T T F Exclusive-Or (or Xor, or OrElse) is similar to this common Or, but it is not often used. The exclusive-Or is true if only one or the other, but not both are true; The Exclusive-Or differs in the last row of the truth table as shown. 6 Logical Views: Control Flow View Behavior of logical operators can be shown by control flow charts as given below. And Or (a & b) (p | q) a p T T b b q q T T T T true false false false true true true false Not x T false true 7 Another View: Lazy Logic (called short circuit): Optional at first Another view of logic is shown.by the following control flow diagrams. Note that if the first condition "a" of the And is false, then the second one "b" is not tested. Note also that if the first condition of the Or is true then the second one is not tested. In other words, sometimes only one condition, the first one, is tested; this is called "short circuit" evaluation or "lazy evaluation". The lazy Or is similar. This “lazy” logic is only slightly faster, and will be treated later, when useful. and or not (a && b) (p || q) (!x) a p x T T T b q T T true false false true true false false true 8 Play Ball: The condition for continuing to play a ball game can be done in many ways: a. In English: play when the score is tied or inning less than or equal to 9 and it's not raining. b. In Symbolic logic: playBall = ( (score1 == score2) | (inning <= 9) ) & ! rain; c. As a control flow chart: two of them; more are possible. s1 == s2 rain T F T tied inn <= 9 inn <= 9 T early T early rain rain s1 == s2 t T tied false true false true false false true true false There are many ways to do anything; there is a tradeoff between logic and program structure. How many tests are needed to prove that this works properly? 3, 4, 5, 7, 8, 12, 16, more? 9 Non-Equality of Real Values Real values (called floats and doubles) should not be compared for equality by the relation "==". This is because real values seem to be very precise, but actually may be approximate. For example, the real value 0.2 when converted to binary computer form is: 0.2 = 0.001100110011 … 0011 … .. forever which continue repeating forever, unending. In practice it is chopped off (often after 32 or 64 bits), and is sufficiently accurate for most needs, but may ultimately cause problems. Comparison between two values, say a, b, can however be made by checking if the difference between them is less than a given small value, such as 0.1 as shown below. isClose = ((a-b) < 0.1) | ((b-a) < 0.1); // within a tenth For closer comparison we can decrease the size of the difference: isVeryClose = ((a-b) < 0.001)| ((b-a) < 0.001); //within a thousanth Also, we can use an absolute value function (from Math class) and specify a "tolerance" of error: TOLERANCE = 0.00001; // Whatever small constant isWithinToleranace = (Math.abs(a-b) < TOLERANCE); 10 Another View of Logic: Data Flow View Behavior of logical operators can be shown by data flow diagrams as given below. For example, a true value flows out of an And only if a true value flows into both inputs.